Average Error: 12.6 → 2.4
Time: 15.6s
Precision: 64
\[\frac{x \cdot \left(y - z\right)}{y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.199818942233228404818074520241924245844 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{y - z}}{x}}\\ \mathbf{elif}\;y \le 2.238050679494542897976277200929792247196 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{x}{\frac{\sqrt{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}}}{\frac{\sqrt{y}}{\sqrt[3]{y - z}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{z}{y}\right) \cdot x\\ \end{array}\]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -2.199818942233228404818074520241924245844 \cdot 10^{-310}:\\
\;\;\;\;\frac{1}{\frac{\frac{y}{y - z}}{x}}\\

\mathbf{elif}\;y \le 2.238050679494542897976277200929792247196 \cdot 10^{-168}:\\
\;\;\;\;\frac{\frac{x}{\frac{\sqrt{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}}}{\frac{\sqrt{y}}{\sqrt[3]{y - z}}}\\

\mathbf{else}:\\
\;\;\;\;\left(1 - \frac{z}{y}\right) \cdot x\\

\end{array}
double f(double x, double y, double z) {
        double r552241 = x;
        double r552242 = y;
        double r552243 = z;
        double r552244 = r552242 - r552243;
        double r552245 = r552241 * r552244;
        double r552246 = r552245 / r552242;
        return r552246;
}
↓
double f(double x, double y, double z) {
        double r552247 = y;
        double r552248 = -2.19981894223323e-310;
        bool r552249 = r552247 <= r552248;
        double r552250 = 1.0;
        double r552251 = z;
        double r552252 = r552247 - r552251;
        double r552253 = r552247 / r552252;
        double r552254 = x;
        double r552255 = r552253 / r552254;
        double r552256 = r552250 / r552255;
        double r552257 = 2.238050679494543e-168;
        bool r552258 = r552247 <= r552257;
        double r552259 = sqrt(r552247);
        double r552260 = cbrt(r552252);
        double r552261 = r552260 * r552260;
        double r552262 = r552259 / r552261;
        double r552263 = r552254 / r552262;
        double r552264 = r552259 / r552260;
        double r552265 = r552263 / r552264;
        double r552266 = r552251 / r552247;
        double r552267 = r552250 - r552266;
        double r552268 = r552267 * r552254;
        double r552269 = r552258 ? r552265 : r552268;
        double r552270 = r552249 ? r552256 : r552269;
        return r552270;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original12.6
Target3.4
Herbie2.4
\[\begin{array}{l} \mathbf{if}\;z \lt -2.060202331921739024383612783691266533098 \cdot 10^{104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z \lt 1.693976601382852594702773997610248441465 \cdot 10^{213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.19981894223323e-310

    1. Initial program 12.4

      \[\frac{x \cdot \left(y - z\right)}{y}\]
    2. Using strategy rm

    3. Applied associate-/l*3.2

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}}\]
    4. Using strategy rm

    5. Applied clear-num3.3

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{y - z}}{x}}}\]

    if -2.19981894223323e-310 < y < 2.238050679494543e-168

    1. Initial program 10.2

      \[\frac{x \cdot \left(y - z\right)}{y}\]
    2. Using strategy rm

    3. Applied associate-/l*12.5

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt13.6

      \[\leadsto \frac{x}{\frac{y}{\color{blue}{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \sqrt[3]{y - z}}}}\]

    6. Applied add-sqr-sqrt13.6

      \[\leadsto \frac{x}{\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\left(\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}\right) \cdot \sqrt[3]{y - z}}}\]

    7. Applied times-frac13.6

      \[\leadsto \frac{x}{\color{blue}{\frac{\sqrt{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}} \cdot \frac{\sqrt{y}}{\sqrt[3]{y - z}}}}\]

    8. Applied associate-/r*1.9

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{\sqrt{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}}}{\frac{\sqrt{y}}{\sqrt[3]{y - z}}}}\]

    if 2.238050679494543e-168 < y

    1. Initial program 13.3

      \[\frac{x \cdot \left(y - z\right)}{y}\]
    2. Using strategy rm

    3. Applied associate-/l*1.2

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}}\]
    4. Using strategy rm

    5. Applied clear-num1.4

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{y - z}}{x}}}\]
    6. Using strategy rm

    7. Applied div-inv1.4

      \[\leadsto \frac{1}{\color{blue}{\frac{y}{y - z} \cdot \frac{1}{x}}}\]

    8. Applied add-cube-cbrt1.4

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{y}{y - z} \cdot \frac{1}{x}}\]

    9. Applied times-frac1.6

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{y}{y - z}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x}}}\]

    10. Simplified1.6

      \[\leadsto \color{blue}{\left(1 - \frac{z}{y}\right)} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x}}\]

    11. Simplified1.4

      \[\leadsto \left(1 - \frac{z}{y}\right) \cdot \color{blue}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.199818942233228404818074520241924245844 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{\frac{y}{y - z}}{x}}\\ \mathbf{elif}\;y \le 2.238050679494542897976277200929792247196 \cdot 10^{-168}:\\ \;\;\;\;\frac{\frac{x}{\frac{\sqrt{y}}{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}}}{\frac{\sqrt{y}}{\sqrt[3]{y - z}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{z}{y}\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e104) (- x (/ (* z x) y)) (if (< z 1.69397660138285259e213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))